Show that if $f \in C^1(\mathbf{R}) \cap CL(\mathbf{R})$ such that $f' \in CL(\mathbf{R})$ and $x \in \mathbf{R}$, then $t \in \mathbf{R}_+ \mapsto P_t(f)(x)$ is of class $C^1$ on $\mathbf{R}_+^*$ and show that for all $t > 0$, we have
$$\frac{\partial P_t(f)(x)}{\partial t} = \int_{-\infty}^{+\infty} \left(-x\mathrm{e}^{-t} + \frac{\mathrm{e}^{-2t}}{\sqrt{1-\mathrm{e}^{-2t}}}\,y\right) f'\!\left(\mathrm{e}^{-t}x + \sqrt{1-\mathrm{e}^{-2t}}\,y\right)\varphi(y)\,\mathrm{d}y.$$